dorsal/arxiv
View SchemaMixed Weyl Symbol Calculus and Spectral Line Shape Theory
| Authors | T. A. Osborn, M. F. Kondrat'eva, G. C. Tabisz, B. R. McQuarrie |
|---|---|
| Categories | |
| ArXiv ID | physics/9811047 |
| URL | https://arxiv.org/abs/physics/9811047 |
| DOI | 10.1088/0305-4470/32/22/315 |
| Journal | Journal of Physics A 32, 4149 (1999). |
Abstract
A new and computationally viable full quantum version of line shape theory is obtained in terms of a mixed Weyl symbol calculus. The basic ingredient in the collision--broadened line shape theory is the time dependent dipole autocorrelation function of the radiator-perturber system. The observed spectral intensity is the Fourier transform of this correlation function. A modified form of the Wigner--Weyl isomorphism between quantum operators and phase space functions (Weyl symbols) is introduced in order to describe the quantum structure of this system. This modification uses a partial Wigner transform in which the radiator-perturber relative motion degrees of freedom are transformed into a phase space dependence, while operators associated with the internal molecular degrees of freedom are kept in their original Hilbert space form. The result of this partial Wigner transform is called a mixed Weyl symbol. The star product, Moyal bracket and asymptotic expansions native to the mixed Weyl symbol calculus are determined. The correlation function is represented as the phase space integral of the product of two mixed symbols: one corresponding to the initial configuration of the system, the other being its time evolving dynamical value. There are, in this approach, two semiclassical expansions -- one associated with the perturber scattering process, the other with the mixed symbol star product. These approximations are used in combination to obtain representations of the autocorrelation that are sufficiently simple to allow numerical calculation. The leading O(\hbar^0) approximation recovers the standard classical path approximation for line shapes. The higher order O(\hbar^1) corrections arise from the noncommutative nature of the star product.
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"abstract": "A new and computationally viable full quantum version of line shape theory is\nobtained in terms of a mixed Weyl symbol calculus. The basic ingredient in the\ncollision--broadened line shape theory is the time dependent dipole\nautocorrelation function of the radiator-perturber system. The observed\nspectral intensity is the Fourier transform of this correlation function. A\nmodified form of the Wigner--Weyl isomorphism between quantum operators and\nphase space functions (Weyl symbols) is introduced in order to describe the\nquantum structure of this system. This modification uses a partial Wigner\ntransform in which the radiator-perturber relative motion degrees of freedom\nare transformed into a phase space dependence, while operators associated with\nthe internal molecular degrees of freedom are kept in their original Hilbert\nspace form. The result of this partial Wigner transform is called a mixed Weyl\nsymbol. The star product, Moyal bracket and asymptotic expansions native to the\nmixed Weyl symbol calculus are determined. The correlation function is\nrepresented as the phase space integral of the product of two mixed symbols:\none corresponding to the initial configuration of the system, the other being\nits time evolving dynamical value. There are, in this approach, two\nsemiclassical expansions -- one associated with the perturber scattering\nprocess, the other with the mixed symbol star product. These approximations are\nused in combination to obtain representations of the autocorrelation that are\nsufficiently simple to allow numerical calculation. The leading O(\\hbar^0)\napproximation recovers the standard classical path approximation for line\nshapes. The higher order O(\\hbar^1) corrections arise from the noncommutative\nnature of the star product.",
"arxiv_id": "physics/9811047",
"authors": [
"T. A. Osborn",
"M. F. Kondrat\u0027eva",
"G. C. Tabisz",
"B. R. McQuarrie"
],
"categories": [
"physics.atom-ph",
"physics.chem-ph"
],
"doi": "10.1088/0305-4470/32/22/315",
"journal_ref": "Journal of Physics A 32, 4149 (1999).",
"title": "Mixed Weyl Symbol Calculus and Spectral Line Shape Theory",
"url": "https://arxiv.org/abs/physics/9811047"
},
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